Triangles and tangents galore

Triangle ABC, inscribed in a circle, has AB = 15 and BC = 25. A tangent to the circle is drawn at B, and a line through A parallel to this tangent intersects \overline{BC} at D. Find DC. This is weird. I tried stuff I know 'bout circles and triangles...(which is pretty close to nil)...but I can't get it.Thanks!:D:D

Re: Triangles and tangents galore

It's a strange problem. There are an infinite number of triangles with those measurements. I used Sketchpad to fix AB and then made a circle, centre B, for the locus of C. Then I constructed the rest to find D. As I move C, D moves around a circle of fixed radius, with a radius consistent with your answer for CD, and centre B. But why?

I seem to have landed up with two problems. (i) Why is the locus for D a circle ? and (ii) what is its radius because once you know BD, you know DC.

??????

EDIT: Oh how come I failed to see this earlier:

(cannot add a diagram to an edited post so I'll put in the next post )

angles EBA BCA and DAB are equal (angle props and parallels)

so

triangles DAB and ACB are similar

so AB/BD = CB/AB

I'll leave the rest to you.

Bob

ps. But why the red circle ? working on that now.

No I'm not. I've done it. BD is fixed so D moves on a circle with that radius.

Last edited by bob bundy (2014-02-04 01:20:33)

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Re: Triangles and tangents galore

Re: Triangles and tangents galore

Hi;

Geogebra allows you to experiment. It allows you to spot patterns and relationships that you would never have seen without it. It combines the best aspects of math and programming into a simple interface.

When I started learning geometry we were warned by our teachers not to pay any attention to the diagram that came with the problem. Do not try to think that AC might equal AB based on my diagram. Well, back then when we wrote on animal hide scrolls and used quill pens the drawings were so bad that they were right. But supposing drawings of geometric figures were made to high accuracy, with scale all accurate, what then? Turns out, the human eye is very good in estimating distances and angles. We were doing that way before anyone ever heard of Euclid.

Because geogebra can measure and draw very accurately it allows the problem solver to use his eyes as well as his brain. Relationships pop out very quickly and a whole new method of solving problems is the result. I daresay that if every schoolkid in the world used geogebra they would not need forums like this one for help with their schoolwork because fully 90% of the problems brought in here can be done using geogebra in a couple of minutes.

In addition, it can assist in the learning of geometry as taught in schools, you know, the old fashioned way. Unbelievable to see people that are 70 or 80 years younger than I being so old fashioned. But if you must learn geometry the same way Euclid thought about it then geogebra can help there too. The whole concept is lumped into the new old concept called Experimental Math, EM for brevity. Welcome to the forum.

Check out some neat uses of the gebra in the Computer Math thread. Try them out for yourself.

In mathematics, you don't understand things. You just get used to them.If it ain't broke, fix it until it is.No great discovery was ever made without a bold guess.